LMFDB - Embedded newform 6300.2.ch.d.4301.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6300,2,Mod(1601,6300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6300, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 3, 0, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6300.1601"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6300.ch (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,-4,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.3057532734$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 32 x^{18} - 124 x^{16} + 5094 x^{14} + 61094 x^{12} + 245850 x^{10} + 420152 x^{8} + \cdots + 15876 $ x^20 - 32x^18 - 124x^16 + 5094x^14 + 61094x^12 + 245850x^10 + 420152x^8 + 466066x^6 + 1736173x^4 - 23778*x^2 + 15876
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			4301.8
Root			$1.08776 + 0.808980i$ of defining polynomial
Character	$\chi$	$=$	6300.4301
Dual form			6300.2.ch.d.1601.8

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.54876 + 2.14507i) q^{7} +(1.40119 + 0.808980i) q^{11} +6.11640i q^{13} +(2.69255 - 4.66363i) q^{17} +(-0.667037 + 0.385114i) q^{19} +(3.23568 - 1.86812i) q^{23} +5.62291i q^{29} +(0.201585 + 0.116385i) q^{31} +(0.237653 + 0.411626i) q^{37} +0.682087 q^{41} +3.64192 q^{43} +(-1.94583 - 3.37027i) q^{47} +(-2.20268 + 6.64441i) q^{49} +(11.1404 + 6.43189i) q^{53} +(5.41271 - 9.37509i) q^{59} +(-3.26629 + 1.88579i) q^{61} +(1.57131 - 2.72158i) q^{67} +13.9116i q^{71} +(-6.42944 - 3.71204i) q^{73} +(0.434792 + 4.25858i) q^{77} +(0.416272 + 0.721004i) q^{79} +0.682087 q^{83} +(5.18151 + 8.97464i) q^{89} +(-13.1201 + 9.47283i) q^{91} -16.8308i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{7} + 6 q^{31} - 2 q^{37} + 20 q^{43} + 4 q^{49} + 6 q^{61} - 28 q^{67} + 6 q^{73} + 2 q^{79} - 40 q^{91}+O(q^{100}) $ 20 * q - 4 * q^7 + 6 * q^31 - 2 * q^37 + 20 * q^43 + 4 * q^49 + 6 * q^61 - 28 * q^67 + 6 * q^73 + 2 * q^79 - 40 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6300\mathbb{Z}\right)^\times$.

$n$	$2801$	$3151$	$3277$	$3601$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	1.54876	+	2.14507i	0.585376	+	0.810762i
$7$	1.54876	+	2.14507i	0.585376	+	0.810762i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.40119	+	0.808980i	0.422476	+	0.243917i	0.696136	−	0.717910i	$-0.254901\pi$
$11$	1.40119	+	0.808980i	0.422476	+	0.243917i	−0.273660	+	0.961826i	$0.588234\pi$
$12$		0			0
$13$			6.11640i			1.69638i	0.529689	+	0.848192i	$0.322309\pi$
$13$			6.11640i			1.69638i	−0.529689	+	0.848192i	$0.677691\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.69255	−	4.66363i	0.653039	−	1.13110i	−0.329342	−	0.944211i	$-0.606827\pi$
$17$	2.69255	−	4.66363i	0.653039	−	1.13110i	0.982382	−	0.186886i	$-0.0598396\pi$
$18$		0			0
$19$	−0.667037	+	0.385114i	−0.153029	+	0.0883512i	−0.574559	−	0.818463i	$-0.694826\pi$
$19$	−0.667037	+	0.385114i	−0.153029	+	0.0883512i	0.421530	+	0.906814i	$0.361493\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.23568	−	1.86812i	0.674686	−	0.389530i	−0.123164	−	0.992386i	$-0.539304\pi$
$23$	3.23568	−	1.86812i	0.674686	−	0.389530i	0.797850	+	0.602856i	$0.205971\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$		0			0
$28$		0			0
$29$			5.62291i			1.04415i	0.852900	+	0.522074i	$0.174841\pi$
$29$			5.62291i			1.04415i	−0.852900	+	0.522074i	$0.825159\pi$
$30$		0			0
$31$	0.201585	+	0.116385i	0.0362057	+	0.0209034i	0.517994	−	0.855384i	$-0.326679\pi$
$31$	0.201585	+	0.116385i	0.0362057	+	0.0209034i	−0.481788	+	0.876288i	$0.660012\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	0.237653	+	0.411626i	0.0390698	+	0.0676709i	0.884899	−	0.465783i	$-0.154227\pi$
$37$	0.237653	+	0.411626i	0.0390698	+	0.0676709i	−0.845829	+	0.533454i	$0.820894\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	0.682087			0.106524			0.0532621	−	0.998581i	$-0.483038\pi$
$41$	0.682087			0.106524			0.0532621	+	0.998581i	$0.483038\pi$
$42$		0			0
$43$	3.64192			0.555388			0.277694	−	0.960670i	$-0.410430\pi$
$43$	3.64192			0.555388			0.277694	+	0.960670i	$0.410430\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.94583	−	3.37027i	−0.283828	−	0.491605i	0.688496	−	0.725240i	$-0.258271\pi$
$47$	−1.94583	−	3.37027i	−0.283828	−	0.491605i	−0.972324	+	0.233635i	$0.924938\pi$
$48$		0			0
$49$	−2.20268	+	6.64441i	−0.314669	+	0.949201i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	11.1404	+	6.43189i	1.53025	+	0.883488i	0.999350	+	0.0360503i	$0.0114776\pi$
$53$	11.1404	+	6.43189i	1.53025	+	0.883488i	0.530895	+	0.847437i	$0.321856\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.41271	−	9.37509i	0.704675	−	1.22053i	−0.262134	−	0.965032i	$-0.584426\pi$
$59$	5.41271	−	9.37509i	0.704675	−	1.22053i	0.966809	−	0.255501i	$-0.0822405\pi$
$60$		0			0
$61$	−3.26629	+	1.88579i	−0.418205	+	0.241451i	−0.694309	−	0.719677i	$-0.744290\pi$
$61$	−3.26629	+	1.88579i	−0.418205	+	0.241451i	0.276104	+	0.961128i	$0.410957\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	1.57131	−	2.72158i	0.191966	−	0.332494i	−0.753936	−	0.656948i	$-0.771847\pi$
$67$	1.57131	−	2.72158i	0.191966	−	0.332494i	0.945902	+	0.324454i	$0.105180\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			13.9116i			1.65100i	0.564403	+	0.825500i	$0.309106\pi$
$71$			13.9116i			1.65100i	−0.564403	+	0.825500i	$0.690894\pi$
$72$		0			0
$73$	−6.42944	−	3.71204i	−0.752510	−	0.434462i	0.0740904	−	0.997252i	$-0.476395\pi$
$73$	−6.42944	−	3.71204i	−0.752510	−	0.434462i	−0.826600	+	0.562790i	$0.809728\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	0.434792	+	4.25858i	0.0495492	+	0.485311i
$78$		0			0
$79$	0.416272	+	0.721004i	0.0468342	+	0.0811193i	0.888492	−	0.458892i	$-0.151753\pi$
$79$	0.416272	+	0.721004i	0.0468342	+	0.0811193i	−0.841658	+	0.540011i	$0.818420\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.682087			0.0748688			0.0374344	−	0.999299i	$-0.488081\pi$
$83$	0.682087			0.0748688			0.0374344	+	0.999299i	$0.488081\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	5.18151	+	8.97464i	0.549239	+	0.951310i	0.998327	+	0.0578220i	$0.0184156\pi$
$89$	5.18151	+	8.97464i	0.549239	+	0.951310i	−0.449088	+	0.893487i	$0.648251\pi$
$90$		0			0
$91$	−13.1201	+	9.47283i	−1.37536	+	0.993023i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	16.8308i		−	1.70891i	−0.519524	−	0.854456i	$-0.673891\pi$
$97$		−	16.8308i		−	1.70891i	0.519524	−	0.854456i	$-0.326109\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.15721	−	5.46846i	0.314155	−	0.544132i	−0.665103	−	0.746752i	$-0.731612\pi$
$101$	3.15721	−	5.46846i	0.314155	−	0.544132i	0.979257	+	0.202620i	$0.0649456\pi$
$102$		0			0
$103$	−6.56213	+	3.78865i	−0.646586	+	0.373307i	−0.787147	−	0.616765i	$-0.788443\pi$
$103$	−6.56213	+	3.78865i	−0.646586	+	0.373307i	0.140561	+	0.990072i	$0.455109\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−11.8071	+	6.81684i	−1.14144	+	0.659009i	−0.946786	−	0.321864i	$-0.895691\pi$
$107$	−11.8071	+	6.81684i	−1.14144	+	0.659009i	−0.194650	+	0.980873i	$0.562357\pi$
$108$		0			0
$109$	7.03461	−	12.1843i	0.673793	−	1.16704i	−0.303027	−	0.952982i	$-0.597997\pi$
$109$	7.03461	−	12.1843i	0.673793	−	1.16704i	0.976820	−	0.214062i	$-0.0686695\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			4.55242i			0.428255i	0.976806	+	0.214128i	$0.0686909\pi$
$113$			4.55242i			0.428255i	−0.976806	+	0.214128i	$0.931309\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	14.1739	−	1.44713i	1.29932	−	0.132658i
$120$		0			0
$121$	−4.19110	−	7.25920i	−0.381009	−	0.659927i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	−3.39541			−0.301294			−0.150647	−	0.988588i	$-0.548136\pi$
$127$	−3.39541			−0.301294			−0.150647	+	0.988588i	$0.548136\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	10.4721	+	18.1382i	0.914952	+	1.58474i	0.806972	+	0.590590i	$0.201105\pi$
$131$	10.4721	+	18.1382i	0.914952	+	1.58474i	0.107979	+	0.994153i	$0.465562\pi$
$132$		0			0
$133$	−1.85918	−	0.834394i	−0.161211	−	0.0723511i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−16.1007	−	9.29576i	−1.37558	−	0.794190i	−0.383955	−	0.923352i	$-0.625438\pi$
$137$	−16.1007	−	9.29576i	−1.37558	−	0.794190i	−0.991624	+	0.129161i	$0.958771\pi$
$138$		0			0
$139$			13.0251i			1.10478i	0.833587	+	0.552388i	$0.186283\pi$
$139$			13.0251i			1.10478i	−0.833587	+	0.552388i	$0.813717\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.94805	+	8.57027i	−0.413776	+	0.716682i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.27299	+	5.35376i	−0.759673	+	0.438597i	−0.829178	−	0.558984i	$-0.811191\pi$
$149$	−9.27299	+	5.35376i	−0.759673	+	0.438597i	0.0695052	+	0.997582i	$0.477858\pi$
$150$		0			0
$151$	−5.18256	+	8.97645i	−0.421751	+	0.730493i	−0.996111	−	0.0881092i	$-0.971918\pi$
$151$	−5.18256	+	8.97645i	−0.421751	+	0.730493i	0.574360	+	0.818603i	$0.305251\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	12.9765	+	7.49197i	1.03564	+	0.597924i	0.918594	−	0.395203i	$-0.129326\pi$
$157$	12.9765	+	7.49197i	1.03564	+	0.597924i	0.117042	+	0.993127i	$0.462659\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	9.01855	+	4.04750i	0.710761	+	0.318988i
$162$		0			0
$163$	−2.20310	−	3.81589i	−0.172560	−	0.298883i	0.766754	−	0.641941i	$-0.221871\pi$
$163$	−2.20310	−	3.81589i	−0.172560	−	0.298883i	−0.939314	+	0.343058i	$0.888537\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−8.49296			−0.657205			−0.328603	−	0.944468i	$-0.606578\pi$
$167$	−8.49296			−0.657205			−0.328603	+	0.944468i	$0.606578\pi$
$168$		0			0
$169$	−24.4103			−1.87772
$170$		0			0
$171$		0			0
$172$		0			0
$173$	0.732940	+	1.26949i	0.0557244	+	0.0965175i	0.892542	−	0.450964i	$-0.148920\pi$
$173$	0.732940	+	1.26949i	0.0557244	+	0.0965175i	−0.836818	+	0.547482i	$0.815587\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−13.0846	−	7.55441i	−0.977991	−	0.564643i	−0.0763281	−	0.997083i	$-0.524320\pi$
$179$	−13.0846	−	7.55441i	−0.977991	−	0.564643i	−0.901663	+	0.432439i	$0.857653\pi$
$180$		0			0
$181$			20.1371i			1.49678i	0.663261	+	0.748388i	$0.269172\pi$
$181$			20.1371i			1.49678i	−0.663261	+	0.748388i	$0.730828\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	7.54557	−	4.35644i	0.551787	−	0.318574i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14.2314	−	8.21652i	1.02975	−	0.594527i	0.112837	−	0.993613i	$-0.464006\pi$
$191$	14.2314	−	8.21652i	1.02975	−	0.594527i	0.916913	+	0.399087i	$0.130673\pi$
$192$		0			0
$193$	−12.4675	+	21.5944i	−0.897431	+	1.55440i	−0.0666645	+	0.997775i	$0.521236\pi$
$193$	−12.4675	+	21.5944i	−0.897431	+	1.55440i	−0.830767	+	0.556621i	$0.812098\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5.28956i		−	0.376865i	−0.982086	−	0.188433i	$-0.939659\pi$
$197$		−	5.28956i		−	0.376865i	0.982086	−	0.188433i	$-0.0603407\pi$
$198$		0			0
$199$	4.91550	+	2.83797i	0.348451	+	0.201178i	0.664003	−	0.747730i	$-0.268856\pi$
$199$	4.91550	+	2.83797i	0.348451	+	0.201178i	−0.315552	+	0.948908i	$0.602190\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−12.0616	+	8.70853i	−0.846555	+	0.611219i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.24620			−0.0862013
$210$		0			0
$211$	0.400950			0.0276025			0.0138013	−	0.999905i	$-0.495607\pi$
$211$	0.400950			0.0276025			0.0138013	+	0.999905i	$0.495607\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	0.0625521	+	0.612668i	0.00424631	+	0.0415906i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	28.5246	+	16.4687i	1.91877	+	1.10780i
$222$		0			0
$223$		−	3.55830i		−	0.238281i	−0.992877	−	0.119141i	$-0.961986\pi$
$223$		−	3.55830i		−	0.238281i	0.992877	−	0.119141i	$-0.0380140\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	13.4956	−	23.3751i	0.895736	−	1.55146i	0.0628447	−	0.998023i	$-0.479983\pi$
$227$	13.4956	−	23.3751i	0.895736	−	1.55146i	0.832891	−	0.553437i	$-0.186684\pi$
$228$		0			0
$229$	−0.914543	+	0.528011i	−0.0604347	+	0.0348920i	−0.529913	−	0.848052i	$-0.677775\pi$
$229$	−0.914543	+	0.528011i	−0.0604347	+	0.0348920i	0.469478	+	0.882944i	$0.344442\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.12617	−	3.53695i	0.401339	−	0.231713i	−0.285723	−	0.958312i	$-0.592234\pi$
$233$	6.12617	−	3.53695i	0.401339	−	0.231713i	0.687062	+	0.726599i	$0.258900\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$			18.3849i			1.18922i	0.804013	+	0.594612i	$0.202694\pi$
$239$			18.3849i			1.18922i	−0.804013	+	0.594612i	$0.797306\pi$
$240$		0			0
$241$	1.41894	+	0.819225i	0.0914019	+	0.0527709i	0.545004	−	0.838433i	$-0.316528\pi$
$241$	1.41894	+	0.819225i	0.0914019	+	0.0527709i	−0.453602	+	0.891204i	$0.649861\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−2.35551	−	4.07986i	−0.149877	−	0.259595i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	11.4769			0.724418			0.362209	−	0.932097i	$-0.382023\pi$
$251$	11.4769			0.724418			0.362209	+	0.932097i	$0.382023\pi$
$252$		0			0
$253$	6.04509			0.380052
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.7381	−	20.3310i	−0.732203	−	1.26821i	−0.955940	−	0.293563i	$-0.905159\pi$
$257$	−11.7381	−	20.3310i	−0.732203	−	1.26821i	0.223737	−	0.974649i	$-0.428174\pi$
$258$		0			0
$259$	−0.514902	+	1.14729i	−0.0319945	+	0.0712893i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	17.3843	+	10.0368i	1.07196	+	0.618897i	0.928716	−	0.370791i	$-0.120913\pi$
$263$	17.3843	+	10.0368i	1.07196	+	0.618897i	0.143244	+	0.989687i	$0.454247\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−9.16917	+	15.8815i	−0.559054	+	0.968311i	0.438521	+	0.898721i	$0.355502\pi$
$269$	−9.16917	+	15.8815i	−0.559054	+	0.968311i	−0.997576	+	0.0695898i	$0.977831\pi$
$270$		0			0
$271$	−18.6007	+	10.7391i	−1.12991	+	0.652356i	−0.943913	−	0.330193i	$-0.892886\pi$
$271$	−18.6007	+	10.7391i	−1.12991	+	0.652356i	−0.186001	+	0.982550i	$0.559553\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4.13213	−	7.15706i	0.248276	−	0.430026i	−0.714772	−	0.699358i	$-0.753469\pi$
$277$	4.13213	−	7.15706i	0.248276	−	0.430026i	0.963047	+	0.269332i	$0.0868028\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			19.9460i			1.18988i	0.803771	+	0.594939i	$0.202824\pi$
$281$			19.9460i			1.18988i	−0.803771	+	0.594939i	$0.797176\pi$
$282$		0			0
$283$	−0.744420	−	0.429791i	−0.0442512	−	0.0255484i	0.477711	−	0.878517i	$-0.341467\pi$
$283$	−0.744420	−	0.429791i	−0.0442512	−	0.0255484i	−0.521962	+	0.852969i	$0.674800\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.05639	+	1.46313i	0.0623567	+	0.0863657i
$288$		0			0
$289$	−5.99964	−	10.3917i	−0.352920	−	0.611275i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	14.0627			0.821549			0.410775	−	0.911737i	$-0.365258\pi$
$293$	14.0627			0.821549			0.410775	+	0.911737i	$0.365258\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	11.4262	+	19.7907i	0.660793	+	1.14453i
$300$		0			0
$301$	5.64046	+	7.81219i	0.325111	+	0.450287i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		−	13.1301i		−	0.749377i	−0.927151	−	0.374689i	$-0.877750\pi$
$307$		−	13.1301i		−	0.749377i	0.927151	−	0.374689i	$-0.122250\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	6.85906	−	11.8802i	0.388942	−	0.673667i	−0.603366	−	0.797465i	$-0.706174\pi$
$311$	6.85906	−	11.8802i	0.388942	−	0.673667i	0.992307	+	0.123798i	$0.0395074\pi$
$312$		0			0
$313$	7.14811	−	4.12696i	0.404035	−	0.233270i	−0.284189	−	0.958768i	$-0.591724\pi$
$313$	7.14811	−	4.12696i	0.404035	−	0.233270i	0.688224	+	0.725499i	$0.258391\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	2.00670	−	1.15857i	0.112707	−	0.0650717i	−0.442587	−	0.896726i	$-0.645939\pi$
$317$	2.00670	−	1.15857i	0.112707	−	0.0650717i	0.555294	+	0.831654i	$0.312606\pi$
$318$		0			0
$319$	−4.54882	+	7.87879i	−0.254685	+	0.441127i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			4.14775i			0.230787i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.21586	−	9.39369i	0.232428	−	0.517891i
$330$		0			0
$331$	9.65316	+	16.7198i	0.530585	+	0.919001i	0.999363	+	0.0356847i	$0.0113612\pi$
$331$	9.65316	+	16.7198i	0.530585	+	0.919001i	−0.468778	+	0.883316i	$0.655305\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	21.8405			1.18973			0.594865	−	0.803826i	$-0.297205\pi$
$337$	21.8405			1.18973			0.594865	+	0.803826i	$0.297205\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	0.188307	+	0.326157i	0.0101974	+	0.0176624i
$342$		0			0
$343$	−17.6642	+	5.56568i	−0.953776	+	0.300518i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	11.8669	+	6.85138i	0.637050	+	0.367801i	0.783477	−	0.621420i	$-0.213444\pi$
$347$	11.8669	+	6.85138i	0.637050	+	0.367801i	−0.146427	+	0.989221i	$0.546777\pi$
$348$		0			0
$349$		−	5.28656i		−	0.282983i	−0.989939	−	0.141491i	$-0.954810\pi$
$349$		−	5.28656i		−	0.282983i	0.989939	−	0.141491i	$-0.0451898\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.06795	−	3.58179i	0.110066	−	0.190640i	−0.805731	−	0.592282i	$-0.798227\pi$
$353$	2.06795	−	3.58179i	0.110066	−	0.190640i	0.915797	+	0.401642i	$0.131561\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	26.7125	−	15.4225i	1.40983	−	0.813967i	0.414460	−	0.910068i	$-0.363971\pi$
$359$	26.7125	−	15.4225i	1.40983	−	0.813967i	0.995372	+	0.0961010i	$0.0306372\pi$
$360$		0			0
$361$	−9.20337	+	15.9407i	−0.484388	+	0.838985i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	4.81917	+	2.78235i	0.251559	+	0.145237i	0.620478	−	0.784224i	$-0.286939\pi$
$367$	4.81917	+	2.78235i	0.251559	+	0.145237i	−0.368919	+	0.929462i	$0.620272\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	3.45686	+	33.8583i	0.179471	+	1.75784i
$372$		0			0
$373$	5.22523	+	9.05036i	0.270552	+	0.468610i	0.969003	−	0.247048i	$-0.0794604\pi$
$373$	5.22523	+	9.05036i	0.270552	+	0.468610i	−0.698451	+	0.715658i	$0.746127\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−34.3919			−1.77127
$378$		0			0
$379$	22.4235			1.15182			0.575910	−	0.817513i	$-0.304648\pi$
$379$	22.4235			1.15182			0.575910	+	0.817513i	$0.304648\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.0641	+	24.3597i	0.718642	+	1.24472i	0.961538	+	0.274672i	$0.0885694\pi$
$383$	14.0641	+	24.3597i	0.718642	+	1.24472i	−0.242896	+	0.970052i	$0.578097\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−3.65189	−	2.10842i	−0.185158	−	0.106901i	0.404556	−	0.914513i	$-0.367426\pi$
$389$	−3.65189	−	2.10842i	−0.185158	−	0.106901i	−0.589714	+	0.807612i	$0.700759\pi$
$390$		0			0
$391$		−	20.1200i		−	1.01751i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.31127	+	0.757060i	−0.0658106	+	0.0379957i	−0.532544	−	0.846402i	$-0.678764\pi$
$397$	−1.31127	+	0.757060i	−0.0658106	+	0.0379957i	0.466734	+	0.884398i	$0.345431\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−28.8743	+	16.6706i	−1.44191	+	0.832490i	−0.997978	−	0.0635679i	$-0.979752\pi$
$401$	−28.8743	+	16.6706i	−1.44191	+	0.832490i	−0.443937	+	0.896058i	$0.646419\pi$
$402$		0			0
$403$	−0.711858	+	1.23297i	−0.0354602	+	0.0614188i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			0.769025i			0.0381191i
$408$		0			0
$409$	2.82101	+	1.62871i	0.139490	+	0.0805346i	0.568121	−	0.822945i	$-0.307671\pi$
$409$	2.82101	+	1.62871i	0.139490	+	0.0805346i	−0.428631	+	0.903480i	$0.641004\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	28.4933	−	2.90910i	1.40206	−	0.143147i
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	38.9059			1.90068			0.950340	−	0.311215i	$-0.100736\pi$
$419$	38.9059			1.90068			0.950340	+	0.311215i	$0.100736\pi$
$420$		0			0
$421$	−37.7183			−1.83828			−0.919139	−	0.393934i	$-0.871114\pi$
$421$	−37.7183			−1.83828			−0.919139	+	0.393934i	$0.871114\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−9.10385	−	4.08579i	−0.440566	−	0.197725i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	11.1785	+	6.45391i	0.538449	+	0.310874i	0.744450	−	0.667678i	$-0.232712\pi$
$431$	11.1785	+	6.45391i	0.538449	+	0.310874i	−0.206001	+	0.978552i	$0.566045\pi$
$432$		0			0
$433$			5.59318i			0.268791i	0.990928	+	0.134396i	$0.0429093\pi$
$433$			5.59318i			0.268791i	−0.990928	+	0.134396i	$0.957091\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.43888	+	2.49221i	−0.0688309	+	0.119219i
$438$		0			0
$439$	−31.5177	+	18.1968i	−1.50426	+	0.868484i	−0.504270	+	0.863546i	$0.668238\pi$
$439$	−31.5177	+	18.1968i	−1.50426	+	0.868484i	−0.999988	+	0.00493767i	$0.998428\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.01870	+	5.20695i	−0.428491	+	0.247390i	−0.698704	−	0.715411i	$-0.746239\pi$
$443$	−9.01870	+	5.20695i	−0.428491	+	0.247390i	0.270212	+	0.962801i	$0.412906\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	2.28525i		−	0.107848i	−0.998545	−	0.0539239i	$-0.982827\pi$
$449$		−	2.28525i		−	0.107848i	0.998545	−	0.0539239i	$-0.0171728\pi$
$450$		0			0
$451$	0.955737	+	0.551795i	0.0450039	+	0.0259830i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−16.4078	−	28.4192i	−0.767527	−	1.32940i	−0.938900	−	0.344189i	$-0.888154\pi$
$457$	−16.4078	−	28.4192i	−0.767527	−	1.32940i	0.171373	−	0.985206i	$-0.445180\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.56922			0.212810			0.106405	−	0.994323i	$-0.466066\pi$
$461$	4.56922			0.212810			0.106405	+	0.994323i	$0.466066\pi$
$462$		0			0
$463$	−29.2197			−1.35795			−0.678977	−	0.734159i	$-0.737577\pi$
$463$	−29.2197			−1.35795			−0.678977	+	0.734159i	$0.737577\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.33872	−	14.4431i	−0.385870	−	0.668347i	0.606019	−	0.795450i	$-0.292765\pi$
$467$	−8.33872	−	14.4431i	−0.385870	−	0.668347i	−0.991889	+	0.127103i	$0.959432\pi$
$468$		0			0
$469$	8.27157	−	0.844510i	0.381946	−	0.0389958i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5.10304	+	2.94624i	0.234638	+	0.135468i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2.36679	+	4.09940i	−0.108141	+	0.187306i	−0.915017	−	0.403415i	$-0.867823\pi$
$479$	−2.36679	+	4.09940i	−0.108141	+	0.187306i	0.806876	+	0.590721i	$0.201157\pi$
$480$		0			0
$481$	−2.51767	+	1.45358i	−0.114796	+	0.0662774i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$	−20.0576	+	34.7408i	−0.908898	+	1.57426i	−0.0933009	+	0.995638i	$0.529742\pi$
$487$	−20.0576	+	34.7408i	−0.908898	+	1.57426i	−0.815597	+	0.578620i	$0.803591\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	17.9082i		−	0.808186i	−0.914718	−	0.404093i	$-0.867587\pi$
$491$		−	17.9082i		−	0.808186i	0.914718	−	0.404093i	$-0.132413\pi$
$492$		0			0
$493$	26.2232	+	15.1400i	1.18103	+	0.681869i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−29.8413	+	21.5457i	−1.33857	+	0.966456i
$498$		0			0
$499$	15.1488	+	26.2384i	0.678152	+	1.17459i	0.975537	+	0.219836i	$0.0705521\pi$
$499$	15.1488	+	26.2384i	0.678152	+	1.17459i	−0.297385	+	0.954758i	$0.596115\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	0.181490			0.00809225			0.00404613	−	0.999992i	$-0.498712\pi$
$503$	0.181490			0.00809225			0.00404613	+	0.999992i	$0.498712\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−16.0302	−	27.7651i	−0.710525	−	1.23067i	−0.964660	−	0.263496i	$-0.915124\pi$
$509$	−16.0302	−	27.7651i	−0.710525	−	1.23067i	0.254136	−	0.967169i	$-0.418209\pi$
$510$		0			0
$511$	−1.99506	−	19.5407i	−0.0882564	−	0.864429i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	6.29655i		−	0.276922i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.8863	+	18.8557i	−0.476939	+	0.826083i	−0.999651	−	0.0264264i	$-0.991587\pi$
$521$	−10.8863	+	18.8557i	−0.476939	+	0.826083i	0.522711	+	0.852510i	$0.324921\pi$
$522$		0			0
$523$	−18.9650	+	10.9494i	−0.829280	+	0.478785i	−0.853606	−	0.520919i	$-0.825589\pi$
$523$	−18.9650	+	10.9494i	−0.829280	+	0.478785i	0.0243264	+	0.999704i	$0.492256\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1.08556	−	0.626746i	0.0472875	−	0.0273015i
$528$		0			0
$529$	−4.52025	+	7.82930i	−0.196533	+	0.340404i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			4.17192i			0.180706i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−8.46159	+	7.52819i	−0.364466	+	0.324262i
$540$		0			0
$541$	−9.53205	−	16.5100i	−0.409815	−	0.709820i	0.585054	−	0.810994i	$-0.301073\pi$
$541$	−9.53205	−	16.5100i	−0.409815	−	0.709820i	−0.994869	+	0.101174i	$0.967740\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	26.7493			1.14372			0.571859	−	0.820352i	$-0.306222\pi$
$547$	26.7493			1.14372			0.571859	+	0.820352i	$0.306222\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.16546	−	3.75068i	−0.0922517	−	0.159785i
$552$		0			0
$553$	−0.901902	+	2.00960i	−0.0383528	+	0.0854567i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	27.8099	+	16.0561i	1.17834	+	0.680318i	0.955631	−	0.294566i	$-0.0951751\pi$
$557$	27.8099	+	16.0561i	1.17834	+	0.680318i	0.222714	+	0.974884i	$0.428508\pi$
$558$		0			0
$559$			22.2754i			0.942151i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	14.5972	−	25.2830i	0.615197	−	1.06555i	−0.375153	−	0.926963i	$-0.622410\pi$
$563$	14.5972	−	25.2830i	0.615197	−	1.06555i	0.990350	−	0.138590i	$-0.0442570\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.84837	+	1.06716i	−0.0774876	+	0.0447375i	−0.538243	−	0.842790i	$-0.680912\pi$
$569$	−1.84837	+	1.06716i	−0.0774876	+	0.0447375i	0.460756	+	0.887527i	$0.347578\pi$
$570$		0			0
$571$	−15.7635	+	27.3032i	−0.659682	+	1.14260i	0.321016	+	0.947074i	$0.395976\pi$
$571$	−15.7635	+	27.3032i	−0.659682	+	1.14260i	−0.980698	+	0.195529i	$0.937358\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	17.3288	+	10.0048i	0.721406	+	0.416504i	0.815270	−	0.579081i	$-0.196589\pi$
$577$	17.3288	+	10.0048i	0.721406	+	0.416504i	−0.0938639	+	0.995585i	$0.529922\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.05639	+	1.46313i	0.0438264	+	0.0607008i
$582$		0			0
$583$	10.4065	+	18.0247i	0.430995	+	0.746505i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5.77666			−0.238428			−0.119214	−	0.992869i	$-0.538037\pi$
$587$	−5.77666			−0.238428			−0.119214	+	0.992869i	$0.538037\pi$
$588$		0			0
$589$	−0.179286			−0.00738736
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6.49041	−	11.2417i	−0.266529	−	0.461642i	0.701434	−	0.712734i	$-0.252543\pi$
$593$	−6.49041	−	11.2417i	−0.266529	−	0.461642i	−0.967963	+	0.251093i	$0.919210\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	26.0753	+	15.0546i	1.06541	+	0.615113i	0.926923	−	0.375251i	$-0.122444\pi$
$599$	26.0753	+	15.0546i	1.06541	+	0.615113i	0.138484	+	0.990365i	$0.455777\pi$
$600$		0			0
$601$		−	31.0797i		−	1.26777i	−0.773428	−	0.633884i	$-0.781460\pi$
$601$		−	31.0797i		−	1.26777i	0.773428	−	0.633884i	$-0.218540\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		0			0
$606$		0			0
$607$	34.8689	−	20.1315i	1.41528	−	0.817114i	0.419404	−	0.907800i	$-0.362239\pi$
$607$	34.8689	−	20.1315i	1.41528	−	0.817114i	0.995880	+	0.0906858i	$0.0289059\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20.6139	−	11.9015i	0.833950	−	0.481482i
$612$		0			0
$613$	−7.76670	+	13.4523i	−0.313694	+	0.543334i	−0.979159	−	0.203095i	$-0.934900\pi$
$613$	−7.76670	+	13.4523i	−0.313694	+	0.543334i	0.665465	+	0.746429i	$0.268233\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	24.1475i		−	0.972144i	−0.873919	−	0.486072i	$-0.838429\pi$
$617$		−	24.1475i		−	0.972144i	0.873919	−	0.486072i	$-0.161571\pi$
$618$		0			0
$619$	−42.7110	−	24.6592i	−1.71670	−	0.991138i	−0.924770	−	0.380526i	$-0.875743\pi$
$619$	−42.7110	−	24.6592i	−1.71670	−	0.991138i	−0.791930	−	0.610611i	$-0.790924\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−11.2263	+	25.0143i	−0.449774	+	1.00218i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.55956			0.102057
$630$		0			0
$631$	25.7293			1.02427			0.512134	−	0.858906i	$-0.328855\pi$
$631$	25.7293			1.02427			0.512134	+	0.858906i	$0.328855\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−40.6399	−	13.4725i	−1.61021	−	0.533800i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	28.1334	+	16.2428i	1.11120	+	0.641554i	0.939141	−	0.343532i	$-0.111623\pi$
$641$	28.1334	+	16.2428i	1.11120	+	0.641554i	0.172063	+	0.985086i	$0.444957\pi$
$642$		0			0
$643$		−	29.7863i		−	1.17466i	−0.809349	−	0.587328i	$-0.800180\pi$
$643$		−	29.7863i		−	1.17466i	0.809349	−	0.587328i	$-0.199820\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16.3002	−	28.2328i	0.640828	−	1.10995i	−0.344420	−	0.938815i	$-0.611925\pi$
$647$	16.3002	−	28.2328i	0.640828	−	1.10995i	0.985248	−	0.171131i	$-0.0547421\pi$
$648$		0			0
$649$	15.1685	−	8.75755i	0.595417	−	0.343764i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.15451	+	0.666558i	−0.0451795	+	0.0260844i	−0.522420	−	0.852689i	$-0.674971\pi$
$653$	−1.15451	+	0.666558i	−0.0451795	+	0.0260844i	0.477240	+	0.878773i	$0.341637\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	42.8917i		−	1.67082i	−0.549626	−	0.835411i	$-0.685230\pi$
$659$		−	42.8917i		−	1.67082i	0.549626	−	0.835411i	$-0.314770\pi$
$660$		0			0
$661$	23.7428	+	13.7079i	0.923486	+	0.533175i	0.884746	−	0.466074i	$-0.154332\pi$
$661$	23.7428	+	13.7079i	0.923486	+	0.533175i	0.0387407	+	0.999249i	$0.487665\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	10.5043	+	18.1939i	0.406727	+	0.704472i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−6.10227			−0.235576
$672$		0			0
$673$	−18.1821			−0.700870			−0.350435	−	0.936587i	$-0.613966\pi$
$673$	−18.1821			−0.700870			−0.350435	+	0.936587i	$0.613966\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.49199	+	2.58420i	0.0573419	+	0.0993190i	0.893271	−	0.449518i	$-0.148404\pi$
$677$	1.49199	+	2.58420i	0.0573419	+	0.0993190i	−0.835930	+	0.548837i	$0.815071\pi$
$678$		0			0
$679$	36.1034	−	26.0669i	1.38552	−	1.00036i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−19.3304	−	11.1604i	−0.739657	−	0.427041i	0.0822877	−	0.996609i	$-0.473777\pi$
$683$	−19.3304	−	11.1604i	−0.739657	−	0.427041i	−0.821945	+	0.569568i	$0.807111\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−39.3400	+	68.1388i	−1.49873	+	2.59588i
$690$		0			0
$691$	8.89912	−	5.13791i	0.338539	−	0.195455i	−0.321087	−	0.947050i	$-0.604048\pi$
$691$	8.89912	−	5.13791i	0.338539	−	0.195455i	0.659626	+	0.751594i	$0.270715\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	1.83655	−	3.18100i	0.0695644	−	0.120489i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	33.7536i		−	1.27486i	−0.770510	−	0.637428i	$-0.779999\pi$
$701$		−	33.7536i		−	1.27486i	0.770510	−	0.637428i	$-0.220001\pi$
$702$		0			0
$703$	−0.317046	−	0.183047i	−0.0119576	−	0.00690373i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	16.6200	−	1.69687i	0.625060	−	0.0638173i
$708$		0			0
$709$	−4.03582	−	6.99025i	−0.151568	−	0.262524i	0.780236	−	0.625485i	$-0.215099\pi$
$709$	−4.03582	−	6.99025i	−0.151568	−	0.262524i	−0.931804	+	0.362961i	$0.881766\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.869686			0.0325700
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	11.6144	+	20.1168i	0.433145	+	0.750229i	0.997142	−	0.0755475i	$-0.0240704\pi$
$719$	11.6144	+	20.1168i	0.433145	+	0.750229i	−0.563997	+	0.825777i	$0.690737\pi$
$720$		0			0
$721$	−18.2901	−	8.20855i	−0.681159	−	0.305702i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$			30.7743i			1.14136i	0.821174	+	0.570678i	$0.193320\pi$
$727$			30.7743i			1.14136i	−0.821174	+	0.570678i	$0.806680\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	9.80605	−	16.9846i	0.362690	−	0.628197i
$732$		0			0
$733$	12.0028	−	6.92980i	0.443332	−	0.255958i	−0.261678	−	0.965155i	$-0.584276\pi$
$733$	12.0028	−	6.92980i	0.443332	−	0.255958i	0.705010	+	0.709197i	$0.250942\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.40341	−	2.54231i	0.162202	−	0.0936472i
$738$		0			0
$739$	−0.182348	+	0.315835i	−0.00670777	+	0.0116182i	−0.869360	−	0.494180i	$-0.835468\pi$
$739$	−0.182348	+	0.315835i	−0.00670777	+	0.0116182i	0.862652	+	0.505798i	$0.168802\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	18.3910i		−	0.674701i	−0.941379	−	0.337351i	$-0.890469\pi$
$743$		−	18.3910i		−	0.674701i	0.941379	−	0.337351i	$-0.109531\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−32.9090	−	14.7695i	−1.20247	−	0.539665i
$750$		0			0
$751$	22.2501	+	38.5383i	0.811918	+	1.40628i	0.911520	+	0.411256i	$0.134910\pi$
$751$	22.2501	+	38.5383i	0.811918	+	1.40628i	−0.0996021	+	0.995027i	$0.531757\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−24.4357			−0.888130			−0.444065	−	0.895994i	$-0.646464\pi$
$757$	−24.4357			−0.888130			−0.444065	+	0.895994i	$0.646464\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−13.2293	−	22.9138i	−0.479561	−	0.830624i	0.520164	−	0.854066i	$-0.325871\pi$
$761$	−13.2293	−	22.9138i	−0.479561	−	0.830624i	−0.999725	+	0.0234421i	$0.992537\pi$
$762$		0			0
$763$	37.0311	−	3.78080i	1.34062	−	0.136874i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	57.3418	+	33.1063i	2.07049	+	1.19540i
$768$		0			0
$769$		−	10.9377i		−	0.394424i	−0.980361	−	0.197212i	$-0.936811\pi$
$769$		−	10.9377i		−	0.394424i	0.980361	−	0.197212i	$-0.0631887\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	8.13738	−	14.0944i	0.292681	−	0.506939i	−0.681762	−	0.731574i	$-0.738786\pi$
$773$	8.13738	−	14.0944i	0.292681	−	0.506939i	0.974443	+	0.224636i	$0.0721192\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.454977	+	0.262681i	−0.0163012	+	0.00941153i
$780$		0			0
$781$	−11.2542	+	19.4928i	−0.402706	+	0.697508i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	11.2119	+	6.47320i	0.399661	+	0.230745i	0.686338	−	0.727283i	$-0.259217\pi$
$787$	11.2119	+	6.47320i	0.399661	+	0.230745i	−0.286676	+	0.958027i	$0.592550\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−9.76527	+	7.05060i	−0.347213	+	0.250691i
$792$		0			0
$793$	−11.5342	−	19.9779i	−0.409593	−	0.709436i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−48.9463			−1.73377			−0.866884	−	0.498510i	$-0.833881\pi$
$797$	−48.9463			−1.73377			−0.866884	+	0.498510i	$0.833881\pi$
$798$		0			0
$799$	−20.9570			−0.741404
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.00594	−	10.4026i	−0.211945	−	0.367099i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−23.4901	−	13.5620i	−0.825869	−	0.476816i	0.0265672	−	0.999647i	$-0.491542\pi$
$809$	−23.4901	−	13.5620i	−0.825869	−	0.476816i	−0.852436	+	0.522831i	$0.824876\pi$
$810$		0			0
$811$			23.7669i			0.834568i	0.908776	+	0.417284i	$0.137018\pi$
$811$			23.7669i			0.834568i	−0.908776	+	0.417284i	$0.862982\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.42930	+	1.40255i	−0.0849903	+	0.0490692i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−21.7449	+	12.5544i	−0.758903	+	0.438153i	−0.828902	−	0.559394i	$-0.811034\pi$
$821$	−21.7449	+	12.5544i	−0.758903	+	0.438153i	0.0699988	+	0.997547i	$0.477700\pi$
$822$		0			0
$823$	27.1139	−	46.9627i	0.945132	−	1.63702i	0.189644	−	0.981853i	$-0.439267\pi$
$823$	27.1139	−	46.9627i	0.945132	−	1.63702i	0.755488	−	0.655163i	$-0.227400\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	33.6990i		−	1.17183i	−0.810373	−	0.585914i	$-0.800735\pi$
$827$		−	33.6990i		−	1.17183i	0.810373	−	0.585914i	$-0.199265\pi$
$828$		0			0
$829$	−35.0353	−	20.2276i	−1.21683	−	0.702535i	−0.252588	−	0.967574i	$-0.581282\pi$
$829$	−35.0353	−	20.2276i	−1.21683	−	0.702535i	−0.964238	+	0.265039i	$0.914615\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	25.0562	+	28.1629i	0.868147	+	0.975787i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	12.6553			0.436909			0.218455	−	0.975847i	$-0.429898\pi$
$839$	12.6553			0.436909			0.218455	+	0.975847i	$0.429898\pi$
$840$		0			0
$841$	−2.61708			−0.0902441
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	9.08051	−	20.2330i	0.312010	−	0.695214i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.53794	+	0.887927i	0.0527197	+	0.0304378i
$852$		0			0
$853$		−	21.0912i		−	0.722150i	−0.932537	−	0.361075i	$-0.882410\pi$
$853$		−	21.0912i		−	0.722150i	0.932537	−	0.361075i	$-0.117590\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.59581	+	2.76402i	−0.0545118	+	0.0944173i	−0.891994	−	0.452048i	$-0.850694\pi$
$857$	−1.59581	+	2.76402i	−0.0545118	+	0.0944173i	0.837482	+	0.546465i	$0.184027\pi$
$858$		0			0
$859$	20.7568	−	11.9839i	0.708212	−	0.408886i	−0.102187	−	0.994765i	$-0.532584\pi$
$859$	20.7568	−	11.9839i	0.708212	−	0.408886i	0.810399	+	0.585879i	$0.199251\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−38.8121	+	22.4082i	−1.32118	+	0.762784i	−0.983917	−	0.178626i	$-0.942835\pi$
$863$	−38.8121	+	22.4082i	−1.32118	+	0.762784i	−0.337263	+	0.941410i	$0.609501\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$			1.34702i			0.0456946i
$870$		0			0
$871$	16.6463	+	9.61073i	0.564038	+	0.325647i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	6.21606	+	10.7665i	0.209901	+	0.363560i	0.951683	−	0.307081i	$-0.0993524\pi$
$877$	6.21606	+	10.7665i	0.209901	+	0.363560i	−0.741782	+	0.670641i	$0.766019\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.2360			−1.01868			−0.509338	−	0.860566i	$-0.670110\pi$
$881$	−30.2360			−1.01868			−0.509338	+	0.860566i	$0.670110\pi$
$882$		0			0
$883$	25.6124			0.861925			0.430963	−	0.902370i	$-0.358174\pi$
$883$	25.6124			0.861925			0.430963	+	0.902370i	$0.358174\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.30210	−	10.9156i	−0.211604	−	0.366508i	0.740613	−	0.671932i	$-0.234535\pi$
$887$	−6.30210	−	10.9156i	−0.211604	−	0.366508i	−0.952217	+	0.305424i	$0.901202\pi$
$888$		0			0
$889$	−5.25868	−	7.28341i	−0.176371	−	0.244278i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.59588	+	1.49873i	0.0868677	+	0.0501531i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−0.654423	+	1.13349i	−0.0218262	+	0.0378041i
$900$		0			0
$901$	59.9919	−	34.6363i	1.99862	−	1.15390i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−21.8001	+	37.7588i	−0.723859	+	1.25376i	0.235582	+	0.971854i	$0.424300\pi$
$907$	−21.8001	+	37.7588i	−0.723859	+	1.25376i	−0.959442	+	0.281907i	$0.909033\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	23.8501i		−	0.790188i	−0.918641	−	0.395094i	$-0.870712\pi$
$911$		−	23.8501i		−	0.790188i	0.918641	−	0.395094i	$-0.129288\pi$
$912$		0			0
$913$	0.955737	+	0.551795i	0.0316303	+	0.0182618i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−22.6890	+	50.5552i	−0.749258	+	1.66948i
$918$		0			0
$919$	19.5865	+	33.9249i	0.646100	+	1.11908i	0.984046	+	0.177913i	$0.0569344\pi$
$919$	19.5865	+	33.9249i	0.646100	+	1.11908i	−0.337946	+	0.941165i	$0.609732\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−85.0887			−2.80073
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.21214	+	2.09950i	0.0397692	+	0.0688822i	0.885225	−	0.465163i	$-0.154004\pi$
$929$	1.21214	+	2.09950i	0.0397692	+	0.0688822i	−0.845456	+	0.534046i	$0.820671\pi$
$930$		0			0
$931$	−1.08958	−	5.28035i	−0.0357096	−	0.173056i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	38.5715i		−	1.26007i	−0.776565	−	0.630037i	$-0.783040\pi$
$937$		−	38.5715i		−	1.26007i	0.776565	−	0.630037i	$-0.216960\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.6354	−	21.8852i	0.411904	−	0.713438i	−0.583194	−	0.812333i	$-0.698197\pi$
$941$	12.6354	−	21.8852i	0.411904	−	0.713438i	0.995098	+	0.0988946i	$0.0315307\pi$
$942$		0			0
$943$	2.20702	−	1.27422i	0.0718703	−	0.0414944i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13.7699	+	7.95004i	−0.447461	+	0.258341i	−0.706757	−	0.707456i	$-0.749843\pi$
$947$	−13.7699	+	7.95004i	−0.447461	+	0.258341i	0.259297	+	0.965798i	$0.416509\pi$
$948$		0			0
$949$	22.7043	−	39.3250i	0.737014	−	1.27654i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			21.1062i			0.683696i	0.939755	+	0.341848i	$0.111053\pi$
$953$			21.1062i			0.683696i	−0.939755	+	0.341848i	$0.888947\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−4.99607	−	48.9342i	−0.161332	−	1.58017i
$960$		0			0
$961$	−15.4729	−	26.7999i	−0.499126	−	0.864512i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−5.22722			−0.168096			−0.0840480	−	0.996462i	$-0.526785\pi$
$967$	−5.22722			−0.168096			−0.0840480	+	0.996462i	$0.526785\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.03879	+	8.72744i	0.161702	+	0.280077i	0.935479	−	0.353381i	$-0.114968\pi$
$971$	5.03879	+	8.72744i	0.161702	+	0.280077i	−0.773777	+	0.633458i	$0.781635\pi$
$972$		0			0
$973$	−27.9398	+	20.1728i	−0.895710	+	0.646709i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	49.3463	+	28.4901i	1.57873	+	0.911479i	0.995037	+	0.0995039i	$0.0317256\pi$
$977$	49.3463	+	28.4901i	1.57873	+	0.911479i	0.583691	+	0.811976i	$0.301608\pi$
$978$		0			0
$979$			16.7670i			0.535874i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	21.2069	−	36.7314i	0.676394	−	1.17155i	−0.299665	−	0.954044i	$-0.596875\pi$
$983$	21.2069	−	36.7314i	0.676394	−	1.17155i	0.976059	−	0.217505i	$-0.0697917\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	11.7841	−	6.80355i	0.374712	−	0.216340i
$990$		0			0
$991$	15.0586	−	26.0822i	0.478352	−	0.828529i	−0.521340	−	0.853349i	$-0.674568\pi$
$991$	15.0586	−	26.0822i	0.478352	−	0.828529i	0.999692	+	0.0248194i	$0.00790108\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−37.2519	−	21.5074i	−1.17978	−	0.681146i	−0.223816	−	0.974631i	$-0.571851\pi$
$997$	−37.2519	−	21.5074i	−1.17978	−	0.681146i	−0.955963	+	0.293486i	$0.905185\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6300.2.ch.d.4301.8	yes	20
3.2	odd	2	inner	6300.2.ch.d.4301.7	yes	20
5.2	odd	4		6300.2.dd.d.4049.4		40
5.3	odd	4		6300.2.dd.d.4049.18		40
5.4	even	2		6300.2.ch.e.4301.4	yes	20
7.5	odd	6	inner	6300.2.ch.d.1601.7	✓	20
15.2	even	4		6300.2.dd.d.4049.3		40
15.8	even	4		6300.2.dd.d.4049.17		40
15.14	odd	2		6300.2.ch.e.4301.3	yes	20
21.5	even	6	inner	6300.2.ch.d.1601.8	yes	20
35.12	even	12		6300.2.dd.d.1349.17		40
35.19	odd	6		6300.2.ch.e.1601.3	yes	20
35.33	even	12		6300.2.dd.d.1349.3		40
105.47	odd	12		6300.2.dd.d.1349.18		40
105.68	odd	12		6300.2.dd.d.1349.4		40
105.89	even	6		6300.2.ch.e.1601.4	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6300.2.ch.d.1601.7	✓	20	7.5	odd	6	inner
6300.2.ch.d.1601.8	yes	20	21.5	even	6	inner
6300.2.ch.d.4301.7	yes	20	3.2	odd	2	inner
6300.2.ch.d.4301.8	yes	20	1.1	even	1	trivial
6300.2.ch.e.1601.3	yes	20	35.19	odd	6
6300.2.ch.e.1601.4	yes	20	105.89	even	6
6300.2.ch.e.4301.3	yes	20	15.14	odd	2
6300.2.ch.e.4301.4	yes	20	5.4	even	2
6300.2.dd.d.1349.3		40	35.33	even	12
6300.2.dd.d.1349.4		40	105.68	odd	12
6300.2.dd.d.1349.17		40	35.12	even	12
6300.2.dd.d.1349.18		40	105.47	odd	12
6300.2.dd.d.4049.3		40	15.2	even	4
6300.2.dd.d.4049.4		40	5.2	odd	4
6300.2.dd.d.4049.17		40	15.8	even	4
6300.2.dd.d.4049.18		40	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6300 = 2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6300.ch (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.54876 + 2.14507i) q^{7} +(1.40119 + 0.808980i) q^{11} +6.11640i q^{13} +(2.69255 - 4.66363i) q^{17} +(-0.667037 + 0.385114i) q^{19} +(3.23568 - 1.86812i) q^{23} +5.62291i q^{29} +(0.201585 + 0.116385i) q^{31} +(0.237653 + 0.411626i) q^{37} +0.682087 q^{41} +3.64192 q^{43} +(-1.94583 - 3.37027i) q^{47} +(-2.20268 + 6.64441i) q^{49} +(11.1404 + 6.43189i) q^{53} +(5.41271 - 9.37509i) q^{59} +(-3.26629 + 1.88579i) q^{61} +(1.57131 - 2.72158i) q^{67} +13.9116i q^{71} +(-6.42944 - 3.71204i) q^{73} +(0.434792 + 4.25858i) q^{77} +(0.416272 + 0.721004i) q^{79} +0.682087 q^{83} +(5.18151 + 8.97464i) q^{89} +(-13.1201 + 9.47283i) q^{91} -16.8308i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{7} + 6 q^{31} - 2 q^{37} + 20 q^{43} + 4 q^{49} + 6 q^{61} - 28 q^{67} + 6 q^{73} + 2 q^{79} - 40 q^{91}+O(q^{100}) \) 20 * q - 4 * q^7 + 6 * q^31 - 2 * q^37 + 20 * q^43 + 4 * q^49 + 6 * q^61 - 28 * q^67 + 6 * q^73 + 2 * q^79 - 40 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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